Given Any Rigid Regular Polygon, Draw a Regular Polygon With Double the Number of Sides Using Only the Rigid Regular Polygon, a Straight Edge and a Pen or Pencil.

A recent Instructables presented a method for drawing a regular octagon using a rigid square, a straight edge and a pencil or pen. This Instructables describes another method for solving this same problem in the field of recreational mathematics. The method presented here can be applied to all regular polygons containing any number sides, not just the four-sided regular polygon. In other words, it provides a method for solving the problem posed in the title to this Instructables.

Step 1 presents the theory behind the proposed method, while the remaining two steps apply this theory to an example which doubles the number of sides of an octagon to form a hexadecagon.

For a regular polygon of n sides, the method relies on the following facts:

1. the vertices of a regular polygon lie on a circle;
2. the sides of a regular polygon are chords of a circle;
3. the n central angles subtended by the sides of a regular polygon are all equal and measure (360/n) degrees;
4. the measure of an inscribed angle made from a side of a regular polygon to a vertex of the polygon is one half the measure of the central angle made by that side of the polygon and is equal to (180/n) degrees;
5. it follows from the last fact that if diagonals are drawn from a common vertex to all the (n – 3) remaining vertices of a regular polygon, then the (n – 4) angles between all such adjacent diagonals are equal to (180/n) degrees;
6. it also follows from fact 4 that the two angles made by the sides of a regular polygon intersecting at the common vertex and the two shortest diagonals emanating from that same vertex of the polygon are (180/n) degrees;
7. the interior angles of a regular polygon are 180 × [1 – (2/n)] degrees;
8. when two regular polygons with n sides share a common side, an equilateral concave polygon with (2n – 2) sides is formed (the resulting polygon is no longer a regular polygon because two of its interior angles are twice that of the other interior angles);
9. the angles opposite the interior angles of the (2n – 2) sided concave polygon that are twice the size of the other interior angles measure {360 – 360 × [1 – (2/n)]} = 720/n degrees (this angle is four times the angle made by adjacent pairs of diagonals drawn from a common vertex on a regular polygon with 2n sides or the angles made by the point of intersection of adjacent sides of such a 2n-sided regular polygon and the shortest diagonal drawn from the point of intersection of the adjacent sides);
10. when two regular polygons with n sides share a common side and a vertex from which all diagonals are drawn on each individual polygon to its remaining (n – 3) vertices, if the diagonals and sides (including the common side) emanating from the common vertex are extended beyond the common vertex so that they lie outside the (2n – 2) sided polygon, then the angles between all such extended lines are all equal to (180/n) degrees;
11. the extended lines described in the previous fact can be regarded as the diagonals of a regular polygon with 2n sides whose center is the common vertex of the two n sided polygons with a common side and vertex.

Some of these facts should become more obvious when the details of the method presented in this Instructables are described in the following two steps which describe a method for drawing a hexadecagon using a rigid octagon, a straight edge and a pencil or pen.

The following steps should be performed:

1. trace two copies of the rigid regular octagon onto a piece of paper so that they share a common side;
2. from one of the common vertices on the shared side draw and extend the lengths of all five diagonals on each individual octagon that emanate from this vertex; (note that three of the diagonals on each individual octagon lie on the same straight line as those on its adjacent polygon);
3. extend the lengths of the lines drawn in the previous sub-step so that the length of the lines drawn from their common point of intersection are greater than the length of the longest diagonal of the individual octagons;
4. extend the lengths of the three sides of the two individual octagons that meet at the common vertex; again, the lengths of the lines should be extended from their common point of intersection so that they are greater than the length of the longest diagonal of the individual octagons.

After carrying out these sub-steps, there should be eight extended lines passing through a common point of intersection; at the common point of intersection, the angle between each pair of adjacent lines is equal to half the 45 degree central angle of the individual octagons, namely 22.5 degrees (the central angle of a hexadecagon).

Marking off equal lengths on the extended diagonals and sides of the octagons drawn from the common vertex of both octagons along both sides of each of the lines emanating from the common vertex provides a set of points that when joined to their nearest neighboring point will result in a hexadecagon. The steps involved are as follows:

1. place one vertex of the rigid octagon at the point of intersection of the extended lines drawn in Step 2;
2. depending on the size of the hexadecagon one wishes to draw, choose another vertex of the rigid octagon and line it up with one of the extended lines drawn in Step 2; in the diagram shown above, the other vertex chosen is that belonging to the longest diagonal of the octagon;
3. mark the point where the second chosen vertex lies on the extended line;
4. repeat these three steps for the remaining 15 lines emanating from the point of intersection of the extended lines drawn in Step 2;
5. using a straight edge join each point marked on the extended lines to its nearest neighboring point.

Carrying out these steps results in drawing a regular hexadecagon.

This procedure works with any regular polygon. In the case of the three-sided polygon, which is an equilateral triangle, after placing two equilateral triangles together so that they possess a common side, one extends the three sides of the two triangles that meet at a common vertex on their common side.

Obviously, with the restriction posed in the title of this Instructables, the longest diagonal of 2n-sided polygon will have a length that is a multiple of twice the length of the side or diameters of the original n-sided polygon. If this restriction is removed and the use of a drawing compass is allowed, then the resulting 2n-sided polygon can have any size. In this situation the first four sub-steps of the current step are not necessary. Instead, one simply places the point of the compass at the point of intersection of the extended lines drawn in Step 2 and marks off two equidistant points on each line emanating from the point of intersection of these lines; then proceed with Sub-step 5 above.